High frequency reconstruction (HFR) is a relatively new technology to enhance the quality of audio and speech coding algorithms. To date it has been introduced for use in speech codecs, such as the wideband AMR coder for 3rd generation cellular systems, and audio coders such as mp3 or AAC, where the traditional waveform codecs are supplemented with the high frequency reconstruction algorithm SBR (resulting in mp3PRO or AAC+SBR).
High frequency reconstruction is a very efficient method to code high frequencies of audio and speech signals. As it cannot perform coding on its own, it is always used in combination with a normal waveform based audio coder (e.g. AAC, mp3) or a speech coder. These are responsible for coding the lower frequencies of the spectrum. The basic idea of high frequency reconstruction is that the higher frequencies are not coded and transmitted, but reconstructed in the decoder based on the lower spectrum with help of some additional parameters (mainly data describing the high frequency spectral envelope of the audio signal) which are transmitted in a low bit rate bit stream, which can be transmitted separately or as ancillary data of the base coder. The additional parameters could also be omitted, but as of today the quality reachable by such an approach will be worse compared to a system using additional parameters.
Especially for Audio Coding, HFR significantly improves the coding efficiency especially in the quality range “sounds good, but is not transparent”. This has two main reasons:                Traditional waveform codecs such as mp3 need to reduce the audio bandwidth for very low bitrates since otherwise the artefact level in the spectrum is getting too high. HFR regenerates those high frequencies at very low cost and with good quality. Since HFR allows a low-cost way to create high frequency components, the audio bandwidth coded by the audio coder can be further reduced, resulting in less artefacts and better worst case behaviour of the total system.        HFR can be used in combination with downsampling in the encoder/upsampling in the decoder. In this frequently used scenario the HFR encoder analyses the full bandwidth audio signal, but the signal fed into the audio coder is sampled down to a lower sampling rate. A typical example is HFR rate at 44.1 kHz, and audio coder rate at 22.05 kHz. Running the audio encoder at a low sampling rate is an advantage, because it is usually more efficient at the lower sampling rate. At the decoding side, the decoded low sample rate audio signal is upsampled and the HFR part is added—thus frequencies up to the original Nyquist frequency can be generated although the audio coder runs at e.g. half the sampling rate.        
A basic parameter for a system using HFR is the so-called cross over frequency (COF), i.e. the frequency where normal waveform coding stops and the HFR frequency range begins. The simplest arrangement is to have the COF at a constant frequency. A more advanced solution that has been introduced already is to dynamically adjust the COF to the characteristics of the signal to be coded.
A main problem with HFR is that an audio signal may contain components in higher frequencies which are difficult to reconstruct with the current HFR method, but could more easily be reproduced by other means, e.g. a waveform coding methods or by synthetic signal generation.
A simple example is coding of a signal only consisting of a sine wave above the COF, FIG. 1. Here the COF is 5.5 kHz. As there is no useful signal available in the low frequencies, the HFR method, based on extrapolating the lowband to obtain a highband, will not generate any signal. Accordingly, the sine wave signal cannot be reconstructed. Other means are needed to code this signal in a useful way. In this simple case, HFR systems providing flexible adjustment of COF can already solve the problem to some extent. If the COF is set above the frequency of the sine wave, the signal can be coded very efficiently using the core coder. This assumes, however, that it is possible to do so, which might not always be the case. As mentioned earlier, one of the main advantages of combining HFR with audio coding is the fact that the core coder can run at half the sampling rate (giving higher compression efficiency). In a realistic scenario, such as a 44.1 kHz system with the core running at 22.05 kHz, such a core coder can only code signals up to around 10.5 kHz. However, apart from that, the problem gets significantly more complicated even for parts of the spectrum within the reach of the core coder when considering more complex signals. Real world signals may e.g. contain audible sine wave-like components at high frequencies within a complex spectrum (e.g. little bells), FIG. 2. Adjusting the COF is not a solution in this case, as most of the gain achieved by the HFR method would diminish by using the core coder for a much larger part of the spectrum.